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Presentation Transcript

Fourier Transforms

Outline

Introduction to reciprocal space

Fourier transformation

Some simple functions

• Area and zero frequency components

• 2- dimensions

Separable

Central slice theorem

Spatial frequencies

Filtering

Modulation Transfer Function

real space reciprocal space

For a complete story see:

Brigham “Fast Fourier Transform”

Here we want to cover the practical aspects of Fourier Transforms.

Define the Fourier Transform as:

There are slight variations on this definition (factors of π and the sign in the exponent), we will revisit these latter, i=√-1.

Also recall that

k is a wave-number and has units that are reciprocal to x:

x -> cm

k -> 2π/cm

So while x describes a position in space, k describes a spatial modulation.

Reciprocal variables are also called conjugate variables.

Another pair of conjugate variables are time and angular frequency.

Conditions for the Fourier Transform to Exist

The sufficient condition for the Fourier transform to exist is that the function g(x) is square integrable,

g(x) may be singular or discontinuous and still have a well defined Fourier transform.

The Fourier transform is complex

The Fourier transform G(k) and the original function g(x) are both in general complex.

The Fourier transform can be written as,

The Fourier transform when g(x) is real

The Fourier transform G(k) has a particularly simple form when g(x) is purely real

So the real part of the Fourier transform reports on the even part of g(x) and the imaginary part on the odd part of g(x).

The Fourier transform of a delta function

The Fourier transform of a delta function should help to convince you that the Fourier transform is quite general (since we can build functions from delta functions).

The delta function picks out the zero frequency value,

k

x

The Fourier transform of a delta function

So it take all spatial frequencies to create a delta function.

The fact that the Fourier transform of a delta function exists shows that the FT is complete.

The basis set of functions (sin and cos) are also orthogonal.

So think of the Fourier transform as picking out the unique spectrum of coefficients (weights) of the sines and cosines.

The Fourier transform of the TopHat Function

Define the TopHat function as,

The Fourier transform is,

which reduces to,

The Fourier transform of a cosine Function

Define the cosine function as,

where k0 is the wave-number of the original function.

The Fourier transform is,

which reduces to,

cosine is real and even, and so the Fourier transform is also real and even. Two delta functions since we can not tell the sign of the spatial frequency.

The Fourier transform of a sine Function

Define the sine function as,

where k0 is the wave-number of the original function.

The Fourier transform is,

which reduces to,

sine is real and odd, and so the Fourier transform is imaginary and odd. Two delta functions since we can not tell the sign of the spatial frequency.

Looking at a cosine or sine alone one can not tell the sense of rotation (only the frequency) but if you have both then the sign is measurable.

Even/odd

if g(x) = g(-x), then G(k) = G(-k)

if g(x) = -g(-x), then G(k) = -G(-k)

Conjugate symmetry

if g(x) is purely real and even, then G(k) is purely real.

if g(x) is purely real and odd, then G(k) is purely imaginary.

if g(x) is purely imaginary and even, then G(k) is purely imaginary.

if g(x) is purely imaginary and odd, then G(k) is purely real.

The Fourier transform of the sign function

The sign function is important in filtering applications, it is defined as,

The FT is calculated by expanding about the origin,

The Fourier transform of the Heaviside function

The Heaviside (or step) function can be explored using the result of the sign function

The FT is then,

Consider the conjugate pair,

what is the FT of

rewrite as,

The new term is not a function of x,

so you pick up a frequency dependent phase shift.

Consider the conjugate pair,

what is the FT of

so the Fourier transform scales inversely with the scaling of g(x).

Also called the energy theorem,

The amount of energy (the weight) of the spectrum is not changed by looking at it in reciprocal space.

In other words, you can make the same measurement in either real or reciprocal space.

Also weight of the zero frequency point corresponds to the total integrated area of the function g(x)

Given a function in reciprocal space G(k) we can return to direct space by the inverse FT,

To show this, recall that

The Fourier transform in 2 dimensions

The Fourier transform can act in any number of dimensions,

It is separable

and the order does not matter.

The equivalence of the zero-frequency rule in 2D is the central slice theorem.

or

So a slice of the 2-D FT that passes through the origin corresponds to the 1 D FT of the projection in real space.

We can change the information content in the image by manipulating the information in reciprocal space.

Weighting function in k-space.

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